Interactive Aesthetic Curve Segments
The Visual Computer manuscript No. (will be inserted by the editor)
Norimasa Yoshida · Takafumi Saito
Interactive Aesthetic Curve Segments
Abstract To meet highly aesthetic requirements in industrial design and styling, we propose a new category of aesthetic curve segments. To achieve these aesthetic requirements, we use curves whose logarithmic curvature histograms(LCH) are represented by straight lines. We call such curves aesthetic curves. We identify the overall shapes of aesthetic curves depending on the slope of LCH α, by imposing specific constraints to the general formula of aesthetic curves. For interactive control, we propose a novel method for drawing an aesthetic curve segment by specifying two endpoints and their tangent vectors. We clarify several characteristics of aesthetic curve segments. Keywords an aesthetic curve segment · logarithmic curvature histogram · the radius of curvature
(a) An aesthetic curve segment
(c) Enlarged evolute of (a)
(b) Approximation of (a)
(d) Enlarged evolute of (b)
Fig. 1 (a) An aesthetic curve segment (α = 2.0) and (b) its approximation by a cubic B´ezier curve segment.
1 Introduction Aesthetic appeal is vital for the market success of industrial products. Since the characteristic lines of a car body, for example, are very important for its aesthetic impact, curves in industrial design and styling need to meet aesthetic requirements. Most curves and surfaces used in conventional CAD systems are based on polynomial or rational parametric forms. However, these curves and surfaces, such as NURBS, are not adequate for highly aesthetic requirements. One of the reasons is the difficulty in controlling the curvature. Though the continuity of curvature can be easily satisfied, it is very hard to Norimasa Yoshida Nihon University, 1-2-1 Izumi-cho Narashino Chiba 275-8575, Japan Tel.+Fax: +81-47-474-2634 E-mail: [email protected] Takafumi Saito Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho Koganei-shi Tokyo 184-8588, Japan Tel.+Fax: +81-42-388-7143 E-mail: [email protected]
control the variation of curvature which dominates the distortion of reflected shapes on curved surfaces. Representation of a circular arc with NURBS is a typical example; achieving constant curvature requires special and unnatural settings of the weights and the knot vector. Fig.1 shows another example, where (a) shows one of the desirable curve segments, and (b) shows an approximation of (a) with a cubic B´ezier curve. Although the shape is simple, the B´ezier curve segment has undesirable curvature changes. Harada et al. have shown that many of the aesthetic curves in artificial objects and the natural world are curves whose logarithmic curvature histograms(LCH to be described in Section 2 ) can be approximated by straight lines [6, 19]. They claimed that two typical aesthetically beautiful curves, the Clothoid curve (also known as a Cornu spiral, Euler Spiral or linarc) and the logarithmic spiral, have the property, where the slopes of their LCH are −1 and 1, respectively. These facts mean that the curves with linear LCH have a potential to meet the
2
highly aesthetic requirements in industrial design. In this paper, we call this category of curves with linear LCH aesthetic curves and denote the slope of the LCH as α. Our goal in this paper is to devise an algorithm that can draw an aesthetic curve segment interactively. Miura presented the general formula of aesthetic curves [12,11], which is defined as the function of arc length. However, drawing aesthetic curves is not trivial. Rather, the exercise poses numerous challenges since the possible ranges of parameters, such as the arc length, are not clearly defined. Moreover, when drawing the curve, we do not know the endpoint of the curve unless we draw the curve, since this process requires numerical integration. This greatly prevents user’s controllability. For practical use, the improvement of controllability of aesthetic curves is indispensable. Furthermore, the overall shape or geometric features of an aesthetic curve with an arbitrary value of α(other than 1 or −1) has not been revealed yet.
Norimasa Yoshida, Takafumi Saito
2 Related Work There are many definitions of fairness of curves. Some of them are the curves with minimum strain energy, the curves that can be drawn with a small number of French curve segments, and the curves whose curvature plots consist of few monotone pieces[3]. See [15] for a collection of definitions. Historically, the curves that approximate elastica, which are idealized thin beams, were pursued [4, 9]. Since an idealized thin beam has the minimum elastic energy, the minimization of a fairness functional is widely used for fairing curves and surfaces. Such fairness functionals include the minimum strain energy and the minimum curvature variation [14, 5].
Other approaches are related to more direct control of curvature or its variation. Nutbourne et al introduced intrinsic splines with which a designer specifies the curvature over the arc length and the curve is calculated Our main contributions are the following: by integrating the curvature plot [16]. Roulier et al. deIdentifying the overall shapes of aesthetic scribed monotonous curvature conditions when two end curves: We identify the overall shapes of aesthetic curves points and their signed curvatures are given [17]. Meek by imposing specific constraints to aesthetic curves so et al. presented a guided clothoid spline passing thorthat they become congruent under similarity transfor- ough given points using clothoid segments, circular arcs, mations. By using these constraints, the formulas of aes- and straight line segments [8]. Higashi et al. proposed thetic curves are derived as a function of arc length or a smooth surface generation method that controls curtangential angle. The tangential angle is the angle be- vature distribution by determining a surface shape from tween the tangent line to the curve and x-axis. We find the evolutes of the four boundary curves [7]. Miura inthat the circle involute is also included in aesthetic curves troduced a unit quaternion integral curve for more direct as α = 2. We clarify the shape change of the spiral and manipulation of its curvature and variation of curvature the behavior of the point of inflection and the point of than B´ezier or B-spline curves [10]. Wang et al. described a shape control method based on sufficient monotone infinite curvature, depending on α. curvature variation conditions for planar B´ezier and BInteractive drawing of an aesthetic curve segment: spline curves [18]. Moll et al. [13] proposed minimal enWe propose a novel method for drawing an aesthetic ergy curves that satisfy endpoint constraints for a path curve segment by specifying two endpoints and their planning of flexible wire. It is somewhat similar to our tangents. When α is specified, our aesthetic curve seg- research in that it finds curves that satisfy endpoint conment has similar controllability to a quadratic B´ezier straints. However, the curves and the method for finding curve segment. Numerical integrations are necessary for a curve segment satisfying endpoint constraints are difdrawing an aesthetic curve segment. However comput- ferent from ours. ing points on an aesthetic curve segment on the screen Harada et al. introduced the logarithmic curvature within the maximum error of 1 × 10−10 actually requires histogram(LCH) to quantitatively find a common propless than several milliseconds. erty of many aesthetic curves [6, 19]. They assumed that Clarifying the characteristics of aesthetic curve the curves are planar and their curvature varies monotsegments: We demonstrate the features of aesthetic onously and showed that the LCH of many of aesthetic curves and their evolutes for various control points and curves in artificial and the natural objects can be approxα. Generally, change of α slightly affects the shape of the imated by straight lines. Such objects included birds’ curve, but drastically alters its evolute. We show that the eggs and butterflies’ wings as well as a Japanese sword position of control points and α dictates whether a curve and the key lines of automobiles. They insist that there segment can be drawn. is a strong correlation between the slope of the straight The rest of this paper is organized as follows. Sec- line in the LCH of a curve and its impression. tion 2 reviews the relevant literature. Section 3 derives the formulas of aesthetic curves and clarifies the overall shapes and characteristics. Section 4 presents a method for interactively drawing a curve segment by specifying three control points. The final two sections present summary and discussions.
The LCH can be interpreted as follows: Let s and ρ be the arc length and the radius of curvature, respectively. When a curve is subdivided into infinitesimal segments such that ∆ρ/ρ is constant, the LCH represents the relationship between ρ and ∆s in a double logarithmic graph. See Fig.2. If we assume that the LCH of a
Interactive Aesthetic Curve Segments
log
Dr/r
is constant
3
Ds Dr/r
the slope:
¿
,+3
Ds
r (1 + Dr / r )
,+1
r
6:8JF7L
LIH
,+/
the radius of curvature
K (
log 1 +
(a) A curve
Dr
/
r)
log
,+.
r
(b) Logarithmic curvature histogram
Fig. 2 (a) A curve and (b) its logarithmic curvature histogram.
*-
*,+20 *,+0 *,+.0
,+.0
,+0
,+20
-
JF E>?>E>B=> DCABG
F;9 5 K ;9
Fig. 3 The aesthetic curve in standard form I.
curve can be represented by a straight line whose slope is α, we obtain, log
∆s = α log ρ + c, ∆ρ/ρ
(1)
where c is a constant [12, 11]. Eq.(1) is the fundamental equation of aesthetic curves.
3 Formulas and Overall Shapes of Aesthetic Curves 3.1 Formulas in Standard Form In this section, we derive the formulas of aesthetic curves in standard form for identifying the overall shapes of aesthetic curves. In order to derive the formula of an aesthetic curve with the slope of LCH α, we consider a reference point Pr on the curve. The reference point can be any point on the curve except at ρ = 0 or ρ = ∞. The following constraints are placed at the reference point. See Fig. 3. – Scaling: ρ = 1 at Pr . – Translation: Pr is placed at the origin. – Rotation: The tangent line to the curve at Pr is parallel to x-axis. Then the standard form can be obtained by transforming an aesthetic curve such that the above constraints are satisfied. Using differentials, Eq.(1) can be modified as ds = ρα−1 ec . dρ
(2)
Let Λ = dρ/ds at the reference point Pr . In this case, Λ = e−c and 0 < Λ < ∞. Using Λ, Eq.(2) can be modified as (α−1)
ρ ds = dρ Λ
.
(3)
The arc length s and the tangential angle θ are set to 0 at the reference point Pr . Integrating Eq.(3) with respect to ρ, we obtain, { 1 if α = 0 Λ log ρ s= (4) 1 α (ρ − 1) otherwise Λα
Solving Eq.(4) with respect to ρ, we obtain { eΛs if α = 0 ρ= 1 (Λαs + 1) α otherwise
(5)
Eq.(5) is the Ces`aro equation of aesthetic curves. A Ces`aro equation is an intrinsic equation that specifies a curve in terms of s and ρ. From the fundamental theorem of the local theory of curves in differential geometry [1], curves satisfying Eq.(5) differs by a rigid motion; that is, if curves c1 and c2 satisfy Eq.(5), there exists an orthogonal linear map R and a vector T such that c2 = R·c1 +T . Using Eq.(3) and the relationship of ds = ρdθ, we obtain dθ ds ρα−2 = = (6) dρ ρdρ Λ By setting θ = 0 at the reference point and integrating Eq.(6) with respect to ρ, we obtain { 1 log ρ if α = 1 θ = ρΛα−1 −1 (7) Λ(α−1) otherwise Since ρ changes from 0 to ∞, s and θ may have a upper or a lower bound depending on α. These bounds are shown in Table 1. Solving Eq.(7) with respect to ρ, we obtain { (eΛθ ) if α = 1 1 ρ= (8) ((α − 1)Λθ + 1) α−1 otherwise The point on the aesthetic curve P (θ) whose tangential angle is θ is defined on the complex plane as { ∫ θ (1+i)Λψ e dψ if α = 1 0 P (θ) = ∫ θ (9) 1 iψ α−1 e dψ otherwise ((α − 1)Λψ + 1) 0 where i is the imaginary unit. The point at θ = 0 defined by Eq.(9) goes through the origin and its tangent vector is [ 1 0 ]T . 1 Substituting Eq.(5) into dθ ds = ρ , and integrating it with respect to s setting θ = 0 when s = 0, we obtain 1 − e−Λs if α = 0 log (Λs+1) if α = 1 Λ (10) θ= 1 (Λαs+1)(1− α ) −1 otherwise Λ(α−1)
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Norimasa Yoshida, Takafumi Saito
Table 1 The upper and lower bounds of s and θ standard form II. s θ UB LB UB LB 1 1 α < 0 − Λα α < 1 Λ(1−α) α=0 α=1 1 1 α>0 − Λα α>1 Λ(1−α)
(a)
α=-1
(d)
α=1
*UB represents the upper bound and LB represents the lower bound.
Then the point on the aesthetic curve C(s) whose arc length is s is defined on the complex plane as ∫ s i(1−e−Λu ) e du if α = 0 0 ∫ s i( log (Λu+1) ) Λ du) if α = 1 e C(s) = (11) 0 ( (Λαu+1)(1−1/α) −1 i ∫ Λ(α−1) s du otherwise e 0
(b)
α=0
(e)
α=2
Eq.(9) and Eq.(11) represent the same curve. The tangential angle θ and the arc length s are related by Eq.(10). (c)
3.2 Standard form I and Overall Shapes To identify the overall shapes of aesthetic curves and see how the curves change depending on the slope of LCH α, we consider aesthetic curves of constant Λ. In Section 3.3, aesthetic curves of arbitrary Λ(≥ 0) is considered. On any aesthetic curves of α 6= 1, there always exists a point such that dρ/ds(= Λ) = 1. Setting such a point as Pr , aesthetic curves can be represented in standard form I. As for the aesthetic curves of α = 1, assuming dρ/ds = 1 restricts the possible shapes since dρ/ds = const. at any point on the curve. However, this assumption is useful for understanding the overall shapes of aesthetic curves. We investigate the behavior of aesthetic curves for θ and s evaluated as a function of ρ where ρ ranges from 0 to ∞. Fig.4 represents the graphs of s with respect to ρ (Eq.(4)) and those of θ with respect to ρ (Eq.(7)) when α = −1, 0, 0.5, 1, 2 and 3. s and θ are either finite or infinite at the points of ρ = 0 or ρ = ∞, which determines the behavior of the curve depending of α, as shown in Table 2. Fig.5 represents the overall shapes of aesthetic curves with various αs using the formulas of tangential angle. By definition, θ = 0 and ρ = 1 at the origin. From Tables 1 and 2, we can figure out the following characteristics of aesthetic curves in standard form I depending on the value of α. The aesthetic curves of α < 0: As θ approaches −∞, the curve spirally converges to the point at ρ = 0. The arc length to that point is infinite. The point of θ = 1 1−α , the upper bound, is the point of inflection because ρ = ∞. Since the arc length to that point is finite, the point of inflection exists (not at infinity). When α = −1, the aesthetic curve is the Clothoid curve.
α=0.5
(f)
α=3
Fig. 4 The graphs of s and θ with respect to ρ are shown when α = −1, 0, 0.5, 1, 2, and 3.
Table 2 Arc length s and tangential angle θ at ρ = 0 and ρ = ∞.
α1
=1
Fig. 7 Two configurations for drawing aesthetic curve segments. (b)
L
=0
L
=0.2
L
L
a
=1 (logarithmic spiral)
=0.02
=0.05
L
=0.5
(c)
L
=1
a
=2 (circle involute)
Fig. 6 Aesthetic curves in standard form II with α = −1, 1, 2.
tangential lines at P0 and P2 . If the triangle P0 P1 P2 is a similar triangle of the triangle Pa Pb Pc , then the curve segment defined by Pa Pb Pc can be drawn by transforming the points on the aesthetic curve in Fig.7(a) on the right such that P0 , P1 and P2 are transformed to Pa , Pb and Pc . Note that there might be a case that a similar triangle is not found. Thus the position of control points and α dictate whether a curve segment can be drawn. The similarity of two triangles can be decided by comparing two pairs of angles. Since θd is the change of the tangential angle from Pa to Pc , θd0 = θd . Now we need to find Λ such that θe = θe0 . This can be done by changing the value of Λ using the bisection method, which will be described shortly. See Fig.6 again to see how aesthetic curves in standard form II changes their shapes depending on Λ. When α > 1, the integration range of 0 to θd (Fig.7(a)) may cause a problem because Λ may become infinity. To avoid this, we use the integration range of −θd to 0 (Fig.7(b)) when α > 1. In this case, the bisection method is used to find Λ such that θf = θf0 . Λ has a upper bound depending on the integration range when α 6= 1. As described in Section 3.3, θ may have a upper or lower bound depending on α. When α < 1,
the integration range of [0, θd ] is used. Since the upper bound of θ is 1/(Λ(1 − α)), θd ≤ 1/(Λ(1 − α)) must hold. Therefore, 0 ≤ Λ ≤ 1/(θd (1 − α)). When α > 1, the integration range of [−θd , 0] is used. Since the lower bound is 1/(Λ(1 − α)), −θd ≥ 1/(Λ(1 − α)) must hold. Therefore, 0 ≤ Λ ≤ 1/(θd (α − 1)). When α = 1, there is no upper bound for Λ, so the bisection method is extended so that Λ(≥ 0) can be arbitrarily large. The pseudo code for the bisection method is shown in Appendix A. For computing the points on aesthetic curves, Eq.(11) (the formula by arc length) is better than Eq.(9)(the formula by tangential angle) especially when the curve segment include the (nearby) point of inflection. At the point of inflection, ρ becomes ∞, which causes Eq.(9) numerically unstable. When the curve segment approaches the shape of a circular arc(when Λ approaches 0), 00 may arise in Eq.(11). In this case, Eq.(9) is better. Therefore, when α 1 × 10−2 , we use Eq.(11). Otherwise, we use Eq.(9). We use an adaptive Gaussian quadrature method for numerical integration.
4.2 Curve Shapes and Drawable Regions Fig.8(a)-(c) show aesthetic curve segments with the same control points but with different αs. It is known that two curves may look identical on the screen, yet reveal significant shape differences when plotted to full scale on a large flatbed plotter [2]. In such a situation, their evolutes (or curvature plots) reveal substantial differences. Though the curve segments in (a)-(c) look similar, their evolute are different substantially. In (d)-(f), we show three aesthetic curve segments with different αs. As the triangle formed by the control points gets closer to an isosceles triangle, aesthetic curve segments with different αs get closer (to a circular arc). The position of control points and α dictates whether a curve can be drawn. Fig.9 shows the drawable regions
Interactive Aesthetic Curve Segments
(a)
a=-2
(b)
a=0
(c)
a=-1
a=-0.2
a=0.5
a=0.5
a=2 (d)
a=-1,0.5,2
7
a=-0.2,0.5,1.2(1)
a
(d)
a
(g)
a
=-3
(b)
a
=-2
(c)
a
(e)
a
=-0.1
(f)
a
=-1
a=2 a=-0.2 a=0.5
a=1.2
a=1.2 (e)
(a)
(f)
=-0.5
=0
a=-0.2,0.5,1.2(2)
Fig. 8 In (a)-(c), aesthetic curve segments with their evolute and radii of curvature are shown. (d)-(f) show three aesthetic curve segments with different αs.
of aesthetic curves with various αs. In (a)-(l), each rectangle is placed with their corners at (±1, ±1). The first control point is placed at (−1, 0) and the third control point is placed at (1, 0). The second control point is moved within the rectangle. If an aesthetic curve segment is drawable, the pixel of the second control point is drawn with white. If not, the point is drawn with black. Since the straight line is not included in aesthetic curves, the curve segment cannot be drawn when the control points are collinear. As shown in Fig.9(g) and (h), there is little restriction for the placement of control points when α is between 0.1 and 1. As α becomes smaller than 0 or gets larger than 1, the drawable regions get smaller. This is because the shape of aesthetic curves gets closer to a circle as α gets smaller than 0 or gets larger than 1 as shown in Fig.5. The second control point can be placed outside the rectangle if it is within the drawable region. The drawable regions of Fig.9 were experimentally constructed so that the maximum error is within 1 × 10−8 . If we allow the drawable region to get slightly smaller, we can decrease the maximum error up to 1 × 10−10 . In this case, the black (not drawable) region gets slightly larger especially when α is around 0. For practical purposes, we can use the maximum error of 1×10−10 . When θd gets very large(close to π), the desired accuracy may not be achieved or the computational efficiency is decreased since the integration range gets relatively large. However, to draw an aesthetic curve segment, it is somewhat unusual to place the second control point such that θd gets larger than π/2.
4.3 Computational Cost The computation time of an aesthetic curve segment varies depending on α as well as the integration range
(j)
=0.1
a
=1.5
(h)
a
=0.2 to 1.0
(k)
a
=2
(i)
(l)
a
=1.1
a
=3
Fig. 9 The drawable regions of aesthetic curve segments with various αs.
and the number of points on the curve computed. In our implementation, the points of an aesthetic curve segment are computed with the constant step of tangential angle of around 0.02 in radian, which is close to 1 deg. More sophisticated computation using both the tangential angle and the arc length is preferable, though. Nevertheless, for drawing a curve segment on the screen, the constant step of tangential angle is satisfactory in most cases. We measured the computation time on a Pentium D 3.2GHz computer. We placed the first and the third control points at the same positions as in Fig.9 and moved the second control point randomly within the drawable region in the blue rectangle using the mouse. The total computation time of an aesthetic curve segment, not including the drawing time, is composed of the time of the bisection method and the time for computing points on the curve. Table 3 shows the total computation time with the maximum error of 1 ×10−10 , the percentage of the bisection method against the total computation time, and the standard deviation of the total computation time. As Table 3 shows, an aesthetic curve segment can be computed within less than several milliseconds. Though numerical integrations are necessary for computing the points on an aesthetic curve segment, our implementation shows that it gives fully interactive control.
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Norimasa Yoshida, Takafumi Saito
Table 3 The Computation time of an aesthetic curve segments(not including drawing). α = −1 α=0 α=1 α=2
TCT (ms) 1.01 1.59 0.96 0.75
PB (%) 32 46 37 29
σ (ms) 0.30 0.76 0.43 0.24
TCT, PB, σ represent the total computation time, the percentage of the bisection method against the TCT, and the standard deviation of the TCT, respectively.
(a)
(b)
(c)
(d)
5 Discussion Aesthetic curves are very fascinating, since they can be considered as a generalization of the Clothoid, the logarithmic spiral, the circle involute, and the circle. This paper has shown the overall shapes of aesthetic curves of arbitrary α. As shown in Fig.5, the overall shapes of aesthetic curves gets closer to the shape of a circle when α gets smaller or larger than 0. Therefore, aesthetic curves are meaningful when α is near 0 for practical use. An aesthetic curve segment of α < 0 can be connected to a straight line segment, since the point of inflection exists. We have presented a method for interactively drawing an aesthetic curve segment by specifying three control points and α. An aesthetic curve segment has the convex hull property, but is not affinely invariant. It is invariant under similarity transformations. When connecting two segments, G1 continuity can be easily achieved by placing the three consecutive control points such that they are collinear. For achieving G2 continuity, more tight restriction is imposed for the placement of the control points. There is some room for improving the computation of an aesthetic curve segment. Eq.(9) and (11) can be directly integrable when α = 1 or 2, and the bisection method might be replaced by a more sophisticated technique. To inspect the geometric quality, we compare the swept surface of an aesthetic curve segment (α = 2) with its approximated surface using reflection lines. For creating the approximated surface, we approximate the aesthetic curve segment by a cubic B´ezier curve segment so that the positions, tangential directions, and curvatures coincide at two endpoints, and then sweep the segment by a line segment. See Fig.1 for the aesthetic curve segment and its approximation by a cubic B´ezier curve segment. Fig.10 shows the swept aesthetic surface (left) and its approximated surface (right). When the two surfaces are rotated about x-axis, the reflection line of the swept aesthetic surface moves monotonously, whereas that of its approximated surface oscillates. Fig.11(a) shows the approximation of Fig.1(a) by a cubic B-spline curve that consists of three cubic B´ezier segments. Uniform B-spline curve segments are created so that they interpolates four points on the aesthetic curve segments and their tangential directions at these points are colinear, and then
Fig. 10 In (a)-(d), the reflection lines of the swept surface(left) of an aesthetic curve segment of α = 2 and its approximated surface(right) by a cubic B´ezier curve are shown. The reflection line of the approximated surface oscillates when the two surfaces are rotated.
(a) Spline curve segments
(b) Enlarged Evolute of (a)
Fig. 11 Approximation of an aesthetic curve segment (Fig. 1(a)) by three B´ezier curve segments. Its enlarged evolute is shown in (b).
converted to B´ezier curve segments. The four points on the aesthetic curve segment are placed so that their arc lengths are separated equidistantly. As shown in Fig.11(b), the approximated curve segments exhibit undesirable curvature change and the reflection line of the swept surface oscillates similarly as in the approximated surface of Fig.10, though the amplitude is smaller. From these results, an aesthetic curve segment has a potential to meet highly aesthetic requirements in industrial design and styling. The application of aesthetic curve segments includes designing 2-dimensional objects as well as 3-dimensional objects. In case of 3-dimensional design, aesthetic curves can be used, for example, as a guide to design the silhouettes of an object. Finding the intrinsic property of aesthetic space curves and surfaces is attractive, though. In fairing of surfaces, our aesthetic curve segment can be used as the constraints for key lines. Another application of our aesthetic curve segment is CAD systems. Since our aesthetic curves can be converted into spline curves, such as B´ezier or B-spline within a specified pre-
Interactive Aesthetic Curve Segments
cision, it is possible to use aesthetic curve segments in many CAD systems.
6 Conclusions This paper has presented a new category of aesthetic curve segment. An aesthetic curve segment can be drawn by specifying two endpoints and their tangents. Thus an aesthetic curve segment has similar controllability to a B´ezier curve segment of degree 2. The radius of curvature of aesthetic curves versus the arc length varies monotonously and smoothly. Therefore, a very high quality curve segment can be easily controlled by three control points and α. Decreasing α makes the curve segment become closer to its control polygon. A circular arc can be represented by placing control points so that they form an isosceles triangle. Although numerical integrations are necessary for computing the curve segment, our implementation showed that the points on aesthetic curves can be computed in less than several milliseconds on the screen within the maximum error of 1 × 10−10 . Aesthetic curves are defined such that the logarithmic curvature histograms of curves are represented by straight lines with slope α. We identified the overall shapes of aesthetic curves by considering their standard form. Future research includes more efficient computation of curve segments, higher accuracy, the connection of several segments, and the extension to surfaces. In order to utilize various kinds of existing shape processing techniques, the approximation of the aesthetic curves with conventional curves, such as NURBS, is also an important subject. We envision that our aesthetic curve segment can be used for designing products as well as in many applications for computer graphics.
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9. Mehlum, E.: Nonlinear splines. In: R.E. Barnhill, R.F. Riesenfeld (eds.) Computer Aided Geometric Design, pp. 173–207. Academic Press, Inc (1974) 10. Miura, K.T.: Unit quaternion integral curve: A new type of fair free-form curves. Computer Aided Geometric Design 17(1), 39–58 (2000) 11. Miura, K.T.: A general equation of aesthetic curves and its self-affinity. In: CAD’06 (International CAD Conferences and Exhibitions) to appear (2006) 12. Miura, K.T., Sone, J., Yamashita, A., Kaneko, T.: Derivation of a general formula of aesthetic curves. In: Proceedings of the Eighth International Conference on Humans and Computers(HC2005), pp. 166–171 (2005) 13. Moll, M., Kavraki, L.E.: Path plannning for minimal energy curves of constant length. In: Proceedings of the 2004 IEEE International Conference on Robotics and Automation, pp. 2826–2831. IEEE (2004) 14. Moreton, H., S´equin, C.: Surface design with minimum energy networks. In: Proceedings of the first ACM Symposium on Solid modeling foundations and CAD/CAM applications. ACM (1991) 15. Moreton, H.P.: Minimum curvature variation curves, networks, and surfaces for fair free-form shape design. Ph.D. thesis, University of California at Berkeley (1992) 16. Nutbourne, A.W., McLellan, P.M., Kensit, R.M.L.: Curvature profiles for plane curves. Computer Aided Design 4(4), 176–184 (1972) 17. Roulier, J., Rnado, T., Piper, B.: Fairness and monotone curvature. In: N.S. Sapidis (ed.) Approximation Theory and Functional Analysis, pp. 177–199. SIAM (91) 18. Wang, Y., Zhao, B., Zhang, L., Xu, J., Wang, K., Wang, S.: Designing fair curves using monotone curvature pieces. Computer Aided Geometric Design 21(5), 515–527 (2004) 19. Yoshimoto, F., Harada, T.: Analysis of the characteristics of curves in natural and factory products. In: Proc. of the Second IASTED International Conference on Visualization, Imaging and Image Processing, pp. 276–281. ACTA Press (2002)
Appendix A: The pseudo code for the bisection method double bisection(double alpha, int maxIteration) { double lmin = 0.0,lmax = 1.0, f; int i = 0, enlarge = 0; if ( alpha == 1.0 ) enlarge = 1; else if ( alpha < 1.0 ) lmax = 1 / (thetaD * (1 - alpha)); else if ( alpha > 1.0 ) lmax = 1 / (-thetaD * (1 - alpha)); Lambda = (lmin + lmax) * 0.5; do { if ( alpha 1.0) ) ) { if ( enlarge ) lmax *= 10.0; Lambda += (lmax - Lambda)*0.5; lmin = pLambda; } else { enlarge = 0; Lambda -= (Lambda-lmin)*0.5; lmax = pLambda; } i++; } while ( i < maxIteration ); return -1;// not found }
Norimasa Yoshida · Takafumi Saito
Interactive Aesthetic Curve Segments
Abstract To meet highly aesthetic requirements in industrial design and styling, we propose a new category of aesthetic curve segments. To achieve these aesthetic requirements, we use curves whose logarithmic curvature histograms(LCH) are represented by straight lines. We call such curves aesthetic curves. We identify the overall shapes of aesthetic curves depending on the slope of LCH α, by imposing specific constraints to the general formula of aesthetic curves. For interactive control, we propose a novel method for drawing an aesthetic curve segment by specifying two endpoints and their tangent vectors. We clarify several characteristics of aesthetic curve segments. Keywords an aesthetic curve segment · logarithmic curvature histogram · the radius of curvature
(a) An aesthetic curve segment
(c) Enlarged evolute of (a)
(b) Approximation of (a)
(d) Enlarged evolute of (b)
Fig. 1 (a) An aesthetic curve segment (α = 2.0) and (b) its approximation by a cubic B´ezier curve segment.
1 Introduction Aesthetic appeal is vital for the market success of industrial products. Since the characteristic lines of a car body, for example, are very important for its aesthetic impact, curves in industrial design and styling need to meet aesthetic requirements. Most curves and surfaces used in conventional CAD systems are based on polynomial or rational parametric forms. However, these curves and surfaces, such as NURBS, are not adequate for highly aesthetic requirements. One of the reasons is the difficulty in controlling the curvature. Though the continuity of curvature can be easily satisfied, it is very hard to Norimasa Yoshida Nihon University, 1-2-1 Izumi-cho Narashino Chiba 275-8575, Japan Tel.+Fax: +81-47-474-2634 E-mail: [email protected] Takafumi Saito Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho Koganei-shi Tokyo 184-8588, Japan Tel.+Fax: +81-42-388-7143 E-mail: [email protected]
control the variation of curvature which dominates the distortion of reflected shapes on curved surfaces. Representation of a circular arc with NURBS is a typical example; achieving constant curvature requires special and unnatural settings of the weights and the knot vector. Fig.1 shows another example, where (a) shows one of the desirable curve segments, and (b) shows an approximation of (a) with a cubic B´ezier curve. Although the shape is simple, the B´ezier curve segment has undesirable curvature changes. Harada et al. have shown that many of the aesthetic curves in artificial objects and the natural world are curves whose logarithmic curvature histograms(LCH to be described in Section 2 ) can be approximated by straight lines [6, 19]. They claimed that two typical aesthetically beautiful curves, the Clothoid curve (also known as a Cornu spiral, Euler Spiral or linarc) and the logarithmic spiral, have the property, where the slopes of their LCH are −1 and 1, respectively. These facts mean that the curves with linear LCH have a potential to meet the
2
highly aesthetic requirements in industrial design. In this paper, we call this category of curves with linear LCH aesthetic curves and denote the slope of the LCH as α. Our goal in this paper is to devise an algorithm that can draw an aesthetic curve segment interactively. Miura presented the general formula of aesthetic curves [12,11], which is defined as the function of arc length. However, drawing aesthetic curves is not trivial. Rather, the exercise poses numerous challenges since the possible ranges of parameters, such as the arc length, are not clearly defined. Moreover, when drawing the curve, we do not know the endpoint of the curve unless we draw the curve, since this process requires numerical integration. This greatly prevents user’s controllability. For practical use, the improvement of controllability of aesthetic curves is indispensable. Furthermore, the overall shape or geometric features of an aesthetic curve with an arbitrary value of α(other than 1 or −1) has not been revealed yet.
Norimasa Yoshida, Takafumi Saito
2 Related Work There are many definitions of fairness of curves. Some of them are the curves with minimum strain energy, the curves that can be drawn with a small number of French curve segments, and the curves whose curvature plots consist of few monotone pieces[3]. See [15] for a collection of definitions. Historically, the curves that approximate elastica, which are idealized thin beams, were pursued [4, 9]. Since an idealized thin beam has the minimum elastic energy, the minimization of a fairness functional is widely used for fairing curves and surfaces. Such fairness functionals include the minimum strain energy and the minimum curvature variation [14, 5].
Other approaches are related to more direct control of curvature or its variation. Nutbourne et al introduced intrinsic splines with which a designer specifies the curvature over the arc length and the curve is calculated Our main contributions are the following: by integrating the curvature plot [16]. Roulier et al. deIdentifying the overall shapes of aesthetic scribed monotonous curvature conditions when two end curves: We identify the overall shapes of aesthetic curves points and their signed curvatures are given [17]. Meek by imposing specific constraints to aesthetic curves so et al. presented a guided clothoid spline passing thorthat they become congruent under similarity transfor- ough given points using clothoid segments, circular arcs, mations. By using these constraints, the formulas of aes- and straight line segments [8]. Higashi et al. proposed thetic curves are derived as a function of arc length or a smooth surface generation method that controls curtangential angle. The tangential angle is the angle be- vature distribution by determining a surface shape from tween the tangent line to the curve and x-axis. We find the evolutes of the four boundary curves [7]. Miura inthat the circle involute is also included in aesthetic curves troduced a unit quaternion integral curve for more direct as α = 2. We clarify the shape change of the spiral and manipulation of its curvature and variation of curvature the behavior of the point of inflection and the point of than B´ezier or B-spline curves [10]. Wang et al. described a shape control method based on sufficient monotone infinite curvature, depending on α. curvature variation conditions for planar B´ezier and BInteractive drawing of an aesthetic curve segment: spline curves [18]. Moll et al. [13] proposed minimal enWe propose a novel method for drawing an aesthetic ergy curves that satisfy endpoint constraints for a path curve segment by specifying two endpoints and their planning of flexible wire. It is somewhat similar to our tangents. When α is specified, our aesthetic curve seg- research in that it finds curves that satisfy endpoint conment has similar controllability to a quadratic B´ezier straints. However, the curves and the method for finding curve segment. Numerical integrations are necessary for a curve segment satisfying endpoint constraints are difdrawing an aesthetic curve segment. However comput- ferent from ours. ing points on an aesthetic curve segment on the screen Harada et al. introduced the logarithmic curvature within the maximum error of 1 × 10−10 actually requires histogram(LCH) to quantitatively find a common propless than several milliseconds. erty of many aesthetic curves [6, 19]. They assumed that Clarifying the characteristics of aesthetic curve the curves are planar and their curvature varies monotsegments: We demonstrate the features of aesthetic onously and showed that the LCH of many of aesthetic curves and their evolutes for various control points and curves in artificial and the natural objects can be approxα. Generally, change of α slightly affects the shape of the imated by straight lines. Such objects included birds’ curve, but drastically alters its evolute. We show that the eggs and butterflies’ wings as well as a Japanese sword position of control points and α dictates whether a curve and the key lines of automobiles. They insist that there segment can be drawn. is a strong correlation between the slope of the straight The rest of this paper is organized as follows. Sec- line in the LCH of a curve and its impression. tion 2 reviews the relevant literature. Section 3 derives the formulas of aesthetic curves and clarifies the overall shapes and characteristics. Section 4 presents a method for interactively drawing a curve segment by specifying three control points. The final two sections present summary and discussions.
The LCH can be interpreted as follows: Let s and ρ be the arc length and the radius of curvature, respectively. When a curve is subdivided into infinitesimal segments such that ∆ρ/ρ is constant, the LCH represents the relationship between ρ and ∆s in a double logarithmic graph. See Fig.2. If we assume that the LCH of a
Interactive Aesthetic Curve Segments
log
Dr/r
is constant
3
Ds Dr/r
the slope:
¿
,+3
Ds
r (1 + Dr / r )
,+1
r
6:8JF7L
LIH
,+/
the radius of curvature
K (
log 1 +
(a) A curve
Dr
/
r)
log
,+.
r
(b) Logarithmic curvature histogram
Fig. 2 (a) A curve and (b) its logarithmic curvature histogram.
*-
*,+20 *,+0 *,+.0
,+.0
,+0
,+20
-
JF E>?>E>B=> DCABG
F;9 5 K ;9
Fig. 3 The aesthetic curve in standard form I.
curve can be represented by a straight line whose slope is α, we obtain, log
∆s = α log ρ + c, ∆ρ/ρ
(1)
where c is a constant [12, 11]. Eq.(1) is the fundamental equation of aesthetic curves.
3 Formulas and Overall Shapes of Aesthetic Curves 3.1 Formulas in Standard Form In this section, we derive the formulas of aesthetic curves in standard form for identifying the overall shapes of aesthetic curves. In order to derive the formula of an aesthetic curve with the slope of LCH α, we consider a reference point Pr on the curve. The reference point can be any point on the curve except at ρ = 0 or ρ = ∞. The following constraints are placed at the reference point. See Fig. 3. – Scaling: ρ = 1 at Pr . – Translation: Pr is placed at the origin. – Rotation: The tangent line to the curve at Pr is parallel to x-axis. Then the standard form can be obtained by transforming an aesthetic curve such that the above constraints are satisfied. Using differentials, Eq.(1) can be modified as ds = ρα−1 ec . dρ
(2)
Let Λ = dρ/ds at the reference point Pr . In this case, Λ = e−c and 0 < Λ < ∞. Using Λ, Eq.(2) can be modified as (α−1)
ρ ds = dρ Λ
.
(3)
The arc length s and the tangential angle θ are set to 0 at the reference point Pr . Integrating Eq.(3) with respect to ρ, we obtain, { 1 if α = 0 Λ log ρ s= (4) 1 α (ρ − 1) otherwise Λα
Solving Eq.(4) with respect to ρ, we obtain { eΛs if α = 0 ρ= 1 (Λαs + 1) α otherwise
(5)
Eq.(5) is the Ces`aro equation of aesthetic curves. A Ces`aro equation is an intrinsic equation that specifies a curve in terms of s and ρ. From the fundamental theorem of the local theory of curves in differential geometry [1], curves satisfying Eq.(5) differs by a rigid motion; that is, if curves c1 and c2 satisfy Eq.(5), there exists an orthogonal linear map R and a vector T such that c2 = R·c1 +T . Using Eq.(3) and the relationship of ds = ρdθ, we obtain dθ ds ρα−2 = = (6) dρ ρdρ Λ By setting θ = 0 at the reference point and integrating Eq.(6) with respect to ρ, we obtain { 1 log ρ if α = 1 θ = ρΛα−1 −1 (7) Λ(α−1) otherwise Since ρ changes from 0 to ∞, s and θ may have a upper or a lower bound depending on α. These bounds are shown in Table 1. Solving Eq.(7) with respect to ρ, we obtain { (eΛθ ) if α = 1 1 ρ= (8) ((α − 1)Λθ + 1) α−1 otherwise The point on the aesthetic curve P (θ) whose tangential angle is θ is defined on the complex plane as { ∫ θ (1+i)Λψ e dψ if α = 1 0 P (θ) = ∫ θ (9) 1 iψ α−1 e dψ otherwise ((α − 1)Λψ + 1) 0 where i is the imaginary unit. The point at θ = 0 defined by Eq.(9) goes through the origin and its tangent vector is [ 1 0 ]T . 1 Substituting Eq.(5) into dθ ds = ρ , and integrating it with respect to s setting θ = 0 when s = 0, we obtain 1 − e−Λs if α = 0 log (Λs+1) if α = 1 Λ (10) θ= 1 (Λαs+1)(1− α ) −1 otherwise Λ(α−1)
4
Norimasa Yoshida, Takafumi Saito
Table 1 The upper and lower bounds of s and θ standard form II. s θ UB LB UB LB 1 1 α < 0 − Λα α < 1 Λ(1−α) α=0 α=1 1 1 α>0 − Λα α>1 Λ(1−α)
(a)
α=-1
(d)
α=1
*UB represents the upper bound and LB represents the lower bound.
Then the point on the aesthetic curve C(s) whose arc length is s is defined on the complex plane as ∫ s i(1−e−Λu ) e du if α = 0 0 ∫ s i( log (Λu+1) ) Λ du) if α = 1 e C(s) = (11) 0 ( (Λαu+1)(1−1/α) −1 i ∫ Λ(α−1) s du otherwise e 0
(b)
α=0
(e)
α=2
Eq.(9) and Eq.(11) represent the same curve. The tangential angle θ and the arc length s are related by Eq.(10). (c)
3.2 Standard form I and Overall Shapes To identify the overall shapes of aesthetic curves and see how the curves change depending on the slope of LCH α, we consider aesthetic curves of constant Λ. In Section 3.3, aesthetic curves of arbitrary Λ(≥ 0) is considered. On any aesthetic curves of α 6= 1, there always exists a point such that dρ/ds(= Λ) = 1. Setting such a point as Pr , aesthetic curves can be represented in standard form I. As for the aesthetic curves of α = 1, assuming dρ/ds = 1 restricts the possible shapes since dρ/ds = const. at any point on the curve. However, this assumption is useful for understanding the overall shapes of aesthetic curves. We investigate the behavior of aesthetic curves for θ and s evaluated as a function of ρ where ρ ranges from 0 to ∞. Fig.4 represents the graphs of s with respect to ρ (Eq.(4)) and those of θ with respect to ρ (Eq.(7)) when α = −1, 0, 0.5, 1, 2 and 3. s and θ are either finite or infinite at the points of ρ = 0 or ρ = ∞, which determines the behavior of the curve depending of α, as shown in Table 2. Fig.5 represents the overall shapes of aesthetic curves with various αs using the formulas of tangential angle. By definition, θ = 0 and ρ = 1 at the origin. From Tables 1 and 2, we can figure out the following characteristics of aesthetic curves in standard form I depending on the value of α. The aesthetic curves of α < 0: As θ approaches −∞, the curve spirally converges to the point at ρ = 0. The arc length to that point is infinite. The point of θ = 1 1−α , the upper bound, is the point of inflection because ρ = ∞. Since the arc length to that point is finite, the point of inflection exists (not at infinity). When α = −1, the aesthetic curve is the Clothoid curve.
α=0.5
(f)
α=3
Fig. 4 The graphs of s and θ with respect to ρ are shown when α = −1, 0, 0.5, 1, 2, and 3.
Table 2 Arc length s and tangential angle θ at ρ = 0 and ρ = ∞.
α1
=1
Fig. 7 Two configurations for drawing aesthetic curve segments. (b)
L
=0
L
=0.2
L
L
a
=1 (logarithmic spiral)
=0.02
=0.05
L
=0.5
(c)
L
=1
a
=2 (circle involute)
Fig. 6 Aesthetic curves in standard form II with α = −1, 1, 2.
tangential lines at P0 and P2 . If the triangle P0 P1 P2 is a similar triangle of the triangle Pa Pb Pc , then the curve segment defined by Pa Pb Pc can be drawn by transforming the points on the aesthetic curve in Fig.7(a) on the right such that P0 , P1 and P2 are transformed to Pa , Pb and Pc . Note that there might be a case that a similar triangle is not found. Thus the position of control points and α dictate whether a curve segment can be drawn. The similarity of two triangles can be decided by comparing two pairs of angles. Since θd is the change of the tangential angle from Pa to Pc , θd0 = θd . Now we need to find Λ such that θe = θe0 . This can be done by changing the value of Λ using the bisection method, which will be described shortly. See Fig.6 again to see how aesthetic curves in standard form II changes their shapes depending on Λ. When α > 1, the integration range of 0 to θd (Fig.7(a)) may cause a problem because Λ may become infinity. To avoid this, we use the integration range of −θd to 0 (Fig.7(b)) when α > 1. In this case, the bisection method is used to find Λ such that θf = θf0 . Λ has a upper bound depending on the integration range when α 6= 1. As described in Section 3.3, θ may have a upper or lower bound depending on α. When α < 1,
the integration range of [0, θd ] is used. Since the upper bound of θ is 1/(Λ(1 − α)), θd ≤ 1/(Λ(1 − α)) must hold. Therefore, 0 ≤ Λ ≤ 1/(θd (1 − α)). When α > 1, the integration range of [−θd , 0] is used. Since the lower bound is 1/(Λ(1 − α)), −θd ≥ 1/(Λ(1 − α)) must hold. Therefore, 0 ≤ Λ ≤ 1/(θd (α − 1)). When α = 1, there is no upper bound for Λ, so the bisection method is extended so that Λ(≥ 0) can be arbitrarily large. The pseudo code for the bisection method is shown in Appendix A. For computing the points on aesthetic curves, Eq.(11) (the formula by arc length) is better than Eq.(9)(the formula by tangential angle) especially when the curve segment include the (nearby) point of inflection. At the point of inflection, ρ becomes ∞, which causes Eq.(9) numerically unstable. When the curve segment approaches the shape of a circular arc(when Λ approaches 0), 00 may arise in Eq.(11). In this case, Eq.(9) is better. Therefore, when α 1 × 10−2 , we use Eq.(11). Otherwise, we use Eq.(9). We use an adaptive Gaussian quadrature method for numerical integration.
4.2 Curve Shapes and Drawable Regions Fig.8(a)-(c) show aesthetic curve segments with the same control points but with different αs. It is known that two curves may look identical on the screen, yet reveal significant shape differences when plotted to full scale on a large flatbed plotter [2]. In such a situation, their evolutes (or curvature plots) reveal substantial differences. Though the curve segments in (a)-(c) look similar, their evolute are different substantially. In (d)-(f), we show three aesthetic curve segments with different αs. As the triangle formed by the control points gets closer to an isosceles triangle, aesthetic curve segments with different αs get closer (to a circular arc). The position of control points and α dictates whether a curve can be drawn. Fig.9 shows the drawable regions
Interactive Aesthetic Curve Segments
(a)
a=-2
(b)
a=0
(c)
a=-1
a=-0.2
a=0.5
a=0.5
a=2 (d)
a=-1,0.5,2
7
a=-0.2,0.5,1.2(1)
a
(d)
a
(g)
a
=-3
(b)
a
=-2
(c)
a
(e)
a
=-0.1
(f)
a
=-1
a=2 a=-0.2 a=0.5
a=1.2
a=1.2 (e)
(a)
(f)
=-0.5
=0
a=-0.2,0.5,1.2(2)
Fig. 8 In (a)-(c), aesthetic curve segments with their evolute and radii of curvature are shown. (d)-(f) show three aesthetic curve segments with different αs.
of aesthetic curves with various αs. In (a)-(l), each rectangle is placed with their corners at (±1, ±1). The first control point is placed at (−1, 0) and the third control point is placed at (1, 0). The second control point is moved within the rectangle. If an aesthetic curve segment is drawable, the pixel of the second control point is drawn with white. If not, the point is drawn with black. Since the straight line is not included in aesthetic curves, the curve segment cannot be drawn when the control points are collinear. As shown in Fig.9(g) and (h), there is little restriction for the placement of control points when α is between 0.1 and 1. As α becomes smaller than 0 or gets larger than 1, the drawable regions get smaller. This is because the shape of aesthetic curves gets closer to a circle as α gets smaller than 0 or gets larger than 1 as shown in Fig.5. The second control point can be placed outside the rectangle if it is within the drawable region. The drawable regions of Fig.9 were experimentally constructed so that the maximum error is within 1 × 10−8 . If we allow the drawable region to get slightly smaller, we can decrease the maximum error up to 1 × 10−10 . In this case, the black (not drawable) region gets slightly larger especially when α is around 0. For practical purposes, we can use the maximum error of 1×10−10 . When θd gets very large(close to π), the desired accuracy may not be achieved or the computational efficiency is decreased since the integration range gets relatively large. However, to draw an aesthetic curve segment, it is somewhat unusual to place the second control point such that θd gets larger than π/2.
4.3 Computational Cost The computation time of an aesthetic curve segment varies depending on α as well as the integration range
(j)
=0.1
a
=1.5
(h)
a
=0.2 to 1.0
(k)
a
=2
(i)
(l)
a
=1.1
a
=3
Fig. 9 The drawable regions of aesthetic curve segments with various αs.
and the number of points on the curve computed. In our implementation, the points of an aesthetic curve segment are computed with the constant step of tangential angle of around 0.02 in radian, which is close to 1 deg. More sophisticated computation using both the tangential angle and the arc length is preferable, though. Nevertheless, for drawing a curve segment on the screen, the constant step of tangential angle is satisfactory in most cases. We measured the computation time on a Pentium D 3.2GHz computer. We placed the first and the third control points at the same positions as in Fig.9 and moved the second control point randomly within the drawable region in the blue rectangle using the mouse. The total computation time of an aesthetic curve segment, not including the drawing time, is composed of the time of the bisection method and the time for computing points on the curve. Table 3 shows the total computation time with the maximum error of 1 ×10−10 , the percentage of the bisection method against the total computation time, and the standard deviation of the total computation time. As Table 3 shows, an aesthetic curve segment can be computed within less than several milliseconds. Though numerical integrations are necessary for computing the points on an aesthetic curve segment, our implementation shows that it gives fully interactive control.
8
Norimasa Yoshida, Takafumi Saito
Table 3 The Computation time of an aesthetic curve segments(not including drawing). α = −1 α=0 α=1 α=2
TCT (ms) 1.01 1.59 0.96 0.75
PB (%) 32 46 37 29
σ (ms) 0.30 0.76 0.43 0.24
TCT, PB, σ represent the total computation time, the percentage of the bisection method against the TCT, and the standard deviation of the TCT, respectively.
(a)
(b)
(c)
(d)
5 Discussion Aesthetic curves are very fascinating, since they can be considered as a generalization of the Clothoid, the logarithmic spiral, the circle involute, and the circle. This paper has shown the overall shapes of aesthetic curves of arbitrary α. As shown in Fig.5, the overall shapes of aesthetic curves gets closer to the shape of a circle when α gets smaller or larger than 0. Therefore, aesthetic curves are meaningful when α is near 0 for practical use. An aesthetic curve segment of α < 0 can be connected to a straight line segment, since the point of inflection exists. We have presented a method for interactively drawing an aesthetic curve segment by specifying three control points and α. An aesthetic curve segment has the convex hull property, but is not affinely invariant. It is invariant under similarity transformations. When connecting two segments, G1 continuity can be easily achieved by placing the three consecutive control points such that they are collinear. For achieving G2 continuity, more tight restriction is imposed for the placement of the control points. There is some room for improving the computation of an aesthetic curve segment. Eq.(9) and (11) can be directly integrable when α = 1 or 2, and the bisection method might be replaced by a more sophisticated technique. To inspect the geometric quality, we compare the swept surface of an aesthetic curve segment (α = 2) with its approximated surface using reflection lines. For creating the approximated surface, we approximate the aesthetic curve segment by a cubic B´ezier curve segment so that the positions, tangential directions, and curvatures coincide at two endpoints, and then sweep the segment by a line segment. See Fig.1 for the aesthetic curve segment and its approximation by a cubic B´ezier curve segment. Fig.10 shows the swept aesthetic surface (left) and its approximated surface (right). When the two surfaces are rotated about x-axis, the reflection line of the swept aesthetic surface moves monotonously, whereas that of its approximated surface oscillates. Fig.11(a) shows the approximation of Fig.1(a) by a cubic B-spline curve that consists of three cubic B´ezier segments. Uniform B-spline curve segments are created so that they interpolates four points on the aesthetic curve segments and their tangential directions at these points are colinear, and then
Fig. 10 In (a)-(d), the reflection lines of the swept surface(left) of an aesthetic curve segment of α = 2 and its approximated surface(right) by a cubic B´ezier curve are shown. The reflection line of the approximated surface oscillates when the two surfaces are rotated.
(a) Spline curve segments
(b) Enlarged Evolute of (a)
Fig. 11 Approximation of an aesthetic curve segment (Fig. 1(a)) by three B´ezier curve segments. Its enlarged evolute is shown in (b).
converted to B´ezier curve segments. The four points on the aesthetic curve segment are placed so that their arc lengths are separated equidistantly. As shown in Fig.11(b), the approximated curve segments exhibit undesirable curvature change and the reflection line of the swept surface oscillates similarly as in the approximated surface of Fig.10, though the amplitude is smaller. From these results, an aesthetic curve segment has a potential to meet highly aesthetic requirements in industrial design and styling. The application of aesthetic curve segments includes designing 2-dimensional objects as well as 3-dimensional objects. In case of 3-dimensional design, aesthetic curves can be used, for example, as a guide to design the silhouettes of an object. Finding the intrinsic property of aesthetic space curves and surfaces is attractive, though. In fairing of surfaces, our aesthetic curve segment can be used as the constraints for key lines. Another application of our aesthetic curve segment is CAD systems. Since our aesthetic curves can be converted into spline curves, such as B´ezier or B-spline within a specified pre-
Interactive Aesthetic Curve Segments
cision, it is possible to use aesthetic curve segments in many CAD systems.
6 Conclusions This paper has presented a new category of aesthetic curve segment. An aesthetic curve segment can be drawn by specifying two endpoints and their tangents. Thus an aesthetic curve segment has similar controllability to a B´ezier curve segment of degree 2. The radius of curvature of aesthetic curves versus the arc length varies monotonously and smoothly. Therefore, a very high quality curve segment can be easily controlled by three control points and α. Decreasing α makes the curve segment become closer to its control polygon. A circular arc can be represented by placing control points so that they form an isosceles triangle. Although numerical integrations are necessary for computing the curve segment, our implementation showed that the points on aesthetic curves can be computed in less than several milliseconds on the screen within the maximum error of 1 × 10−10 . Aesthetic curves are defined such that the logarithmic curvature histograms of curves are represented by straight lines with slope α. We identified the overall shapes of aesthetic curves by considering their standard form. Future research includes more efficient computation of curve segments, higher accuracy, the connection of several segments, and the extension to surfaces. In order to utilize various kinds of existing shape processing techniques, the approximation of the aesthetic curves with conventional curves, such as NURBS, is also an important subject. We envision that our aesthetic curve segment can be used for designing products as well as in many applications for computer graphics.
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Appendix A: The pseudo code for the bisection method double bisection(double alpha, int maxIteration) { double lmin = 0.0,lmax = 1.0, f; int i = 0, enlarge = 0; if ( alpha == 1.0 ) enlarge = 1; else if ( alpha < 1.0 ) lmax = 1 / (thetaD * (1 - alpha)); else if ( alpha > 1.0 ) lmax = 1 / (-thetaD * (1 - alpha)); Lambda = (lmin + lmax) * 0.5; do { if ( alpha 1.0) ) ) { if ( enlarge ) lmax *= 10.0; Lambda += (lmax - Lambda)*0.5; lmin = pLambda; } else { enlarge = 0; Lambda -= (Lambda-lmin)*0.5; lmax = pLambda; } i++; } while ( i < maxIteration ); return -1;// not found }
